Will using quads affect game performance?
Stick to quads for modelling, as that is what most modelling tools are designed for. 
Either the game engine should handle triangulating the model, or it should be triangulated just before export from blender if that is not the case. 

The only exception to this is very low polycount assests where optimising the use of triangles is important (to get the right shape with fewer polys, not many games are going to need much of this though). In this case, you will probably want to switch to modelling directly with triangles at some point during the modelling process.

Its worth noting that everything is actually triangulated when rendering anyway,  even for the viewport opengl render while you are modelling. It is just easier to use a quad representation for mesh manipulation.